Question

If the vectors $\overrightarrow{a}=\left(2,{\log }_{3}x,a\right)$ $and\overrightarrow{b}=\left(-3,a{\log }_{3}x,{\log }_{3}x\right)$ are included at an acute angle then-

• A. a=0
• B. a<0
• C. a>0
• D. None of these

Answer 1

The correct option is C a>0

$\begin{array}{c}\overrightarrow{a}=2\left({\log }_{3}x,a\right),\overrightarrow{b}=\left(-3,a{\log }_{3}x,{\log }_{3}x\right)\\ \overrightarrow{a}\overrightarrow{b}=\left|a\right|\left|b\right|\cos \theta \\ \cos \theta =\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|a\right|\left|b\right|}\\ =\frac{-6+a{\left({\log }_{3}x\right)}^2+a.{\log }_{3}x}{\sqrt{4+{\left({\log }_{3}x\right)}^2+a^2}.\sqrt{9+a^2{\left({\log }_{3}x\right)}^2+{\left({\log }_{3}x\right)}^2}}\\ foracuteanglea>0\\ OptionCiscorrect.\end{array}$